Ten years after the discovery of metallocarboranes, this field of chemistry is still advancing at a rapid pace. While the polyhedral expansion of carboranes has proved to be a general method for the synthesis of metallocarboranes, polymetallic compounds may be synthesized by polyhedral expansion of monometallocarboranes, sometimes producing supraicosahedral complexes. The polyhedral contraction...

We introduce a new variety of flexatube, a rhomboflexatube. It is obtained from a cardboard rhombohedron by removing a pair of opposite faces (rhombi), and then subdividing the remaining four faces by pairs of diagonals. It is reversible, that is, it can be turned inside out by a series of folds, using edges and diagonals of the rhombi. To turn a rhomboflexatube inside out is quite a challengin...

We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on c...

horizon map Representing the Visibility Structure of a Polyhedral Terrain through a Horizon Map We present a model for describing the visibility of a polyhedral terrain from a xed viewpoint, based on a collection of nested horizons. We brie y introduce the concepts of mathematical and digital terrain models, and some background notions for visibility problems on terrains. Then, we de ne horizon...

We explore which polyhedra and polyhedral complexes can be formed by folding up a planar polygonal region and fastening it with one zipper. We call the reverse process a zipper unfolding. A zipper unfolding of a polyhedron is a path cut that unfolds the polyhedron to a planar polygon; in the case of edge cuts, these are Hamiltonian unfoldings as introduced by Shephard in 1975. We show that all ...

Polyhedral analysis infers invariant linear equalities and inequalities of imperative programs. However, the exponential complexity of polyhedral operations such as image computation and convex hull limits the applicability of polyhedral analysis. Weakly relational domains such as intervals and octagons address the scalability issue by considering polyhedra whose constraints are drawn from a re...

We consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures. Thereby we place emphasis on dynamic models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favora...

The Polyhedral Model is one of the most powerful framework for automatic optimization and parallelization of high-level programs. It is based on an algebraic representation of program parts and allows to achieve exact data dependence analysis and to apply complex sequences of optimizations seamlessly. After more than twenty years of research and development, this model is now quite mature and r...

Disordered structures such as liquids and glasses, grains and foams, galaxies, etc. are often represented as polyhedral tilings. Characterizing the associated polyhedral tiling is a promising strategy to understand the disordered structure. However, since a variety of polyhedra are arranged in complex ways, it is challenging to describe what polyhedra are tiled in what way. Here, to solve this ...

Casting is a manufacturing process in which molten material is poured into a cast (mould), which is opened after the material has solidified. As in all applications of robotics, we have to deal with imperfect control of the casting machinery. In this paper, we consider directional uncertainty: given a 3-dimensional polyhedral object, is there a polyhedral cast such that its two parts can be rem...